Question

several years ago, the mean height of women 20 years of age or older was 63.7 inches. suppose that a random sample of 45 women who are 20 years of age or older today results in a mean height of 64.1 inches.
(a) state the appropriate null and alternative hypotheses to assess whether women are taller today
(b) suppose the p - value for this test is 0.13. explain what this value represents.
(c) write a conclusion for this hypothesis test assuming an \\( \alpha = 0.05 \\) level of significance
(a) state the appropriate null and alternative hypotheses to assess whether women are taller today
a. \\( h _ { 0 } : \mu = 63.7 \\) in versus \\( h _ { 1 } : \mu \
eq 63.7 \\) in
b. \\( h _ { 0 } : \mu = 64.1 \\) in versus \\( h _ { 1 } : \mu > 64.1 \\) in
c. \\( h _ { 0 } : \mu = 64.1 \\) in versus \\( h _ { 1 } : \mu < 64.1 \\) in
d. \\( h _ { 0 } : \mu = 63.7 \\) in versus \\( h _ { 1 } : \mu > 63.7 \\) in
e. \\( h _ { 0 } : \mu = 64.1 \\) in versus \\( h _ { 1 } : \mu \
eq 64.1 \\) in
f. \\( h _ { 0 } : \mu = 63.7 \\) in versus \\( h _ { 1 } : \mu < 63.7 \\) in
(b) suppose the p - value for this test is 0.13. explain what this value represents
a. there is a 0.13 probability of obtaining a sample mean height of 64.1 inches or taller from a population whose mean height is 63.7 inches.
b. there is a 0.13 probability of obtaining a sample mean height of 64.1 inches or shorter from a population whose mean height is 63.7 inches.
c. there is a 0.13 probability of obtaining a sample mean height of 63.7 inches or taller from a population whose mean height is 64.1 inches.
d. there is a 0.13 probability of obtaining a sample mean height of exactly 64.1 inches from a population whose mean height is 63.7 inches.

Explanation:

• Part (a):
• The null hypothesis \(H_0\) is a statement of no - change or equality. Here, we are comparing the current mean height to the historical mean height. The historical mean height is \(63.7\) inches. The alternative hypothesis \(H_1\) for the claim that women are taller today is a one - sided (right - tailed) test. So \(H_0:\mu = 63.7\) in (no change from the historical mean) and \(H_1:\mu>63.7\) in (women are taller today).
• Part (b):
• The P - value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. In the context of a one - sample mean test (where we are testing if the mean has increased), if the null hypothesis \(H_0:\mu = 63.7\) in is true, the P - value of \(0.13\) represents the probability of getting a sample mean of \(64.1\) inches or larger.

Answer:

• (a) D. \(H_0:\mu = 63.7\) in versus \(H_1:\mu>63.7\) in
• (b) A. There is a \(0.13\) probability of obtaining a sample mean height of \(64.1\) inches or taller from a population whose mean height is \(63.7\) inches.